Maximum confidence has been considered extensively in theory and experiment for solving various quantum information problems[1]}, [2]}, [3]}, [4]}. The strategy aims to discriminate between quantum states with maximum confidence in each conclusive
result, thereby keeping the probability of
inconclusive results as small as possible. The discrimination strategy proceeds as follows. We assume that we have been provided two mixed states \(\rho _{0}\)  and \(\rho _{1}\) , namely the states given in Eq. (REF ), which occur with some prior probability \(\eta _{0}\)  and \(\eta _{1}\)  respectively (here \(\eta _{0}+\eta _{1}=1\) ). A complete measurement strategy is then described by three POVM operators \(\Pi _{0}\) , \(\Pi _{1}\)  and \(\Pi _{?}\) , which sum up to the identity operator \(I_{d}\)  in the \(d\) -dimensional joint Hilbert space \(\mathcal {H}_{d}\) . If \(\Pi _{0}\)  `clicks' we say that the state is \(\rho _0\) , if \(\Pi _{1}\)  `clicks' we say we have \(\rho _{1}\) , and if \(\Pi _{?}\)  clicks we say we have an inconclusive result. Our objective then is find the maximum confidence \(C_{j}\) . This is the conditional probability \(P(\rho _{j}|j)\)  that the state \(\rho _j\)  was indeed prepared give that the detector \(\Pi _j\)  clicked, that is,
\(C_{j}=\frac{\eta _{j}\textrm {Tr}(\rho _{j}\Pi _{j})}{\textrm {Tr}(\rho \Pi _{j})},\) 
